Stronger Methods of Making Quantum Interactive Proofs Perfectly Complete

Abstract

This paper presents stronger methods of achieving perfect completeness in quantum interactive proofs. It is proved that any problem in QMA has a two-message quantum interactive proof system with perfect completeness and constant soundness error, where the verifier has only to send a constant number of halves of EPR pairs. This in particular implies that the class QMA is necessarily included by the class ${{QIP}_1(2)}$ of problems having two-message quantum interactive proofs with perfect completeness, which gives the first nontrivial upper bound for QMA in terms of quantum interactive proofs. It is also proved that any problem having an m-message quantum interactive proof system necessarily has an (m+1)-message quantum interactive proof system with perfect completeness for every ${m \geq 2}$. This improves the previous construction due to Kitaev and Watrous, which increases the number of messages by two to achieve perfect completeness, if not using the parallelization result.